Dissipative Partial Differential Equations and Dynamical Systems
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Dissipative Mechanics Using Complex-Valued Hamiltonians
Dissipative Mechanics Using Complex-Valued Hamiltonians S. G. Rajeev Department of Physics and Astronomy University of Rochester, Rochester, New York 14627 March 18, 2018 Abstract We show that a large class of dissipative systems can be brought to a canonical form by introducing complex co-ordinates in phase space and a complex-valued hamiltonian. A naive canonical quantization of these sys- tems lead to non-hermitean hamiltonian operators. The excited states are arXiv:quant-ph/0701141v1 19 Jan 2007 unstable and decay to the ground state . We also compute the tunneling amplitude across a potential barrier. 1 1 Introduction In many physical situations, loss of energy of the system under study to the out- side environment cannot be ignored. Often, the long time behavior of the system is determined by this loss of energy, leading to interesting phenomena such as attractors. There is an extensive literature on dissipative systems at both the classical and quantum levels (See for example the textbooks [1, 2, 3]). Often the theory is based on an evolution equation of the density matrix of a ‘small system’ coupled to a ‘reservoir’ with a large number of degrees of freedom, after the reservoir has been averaged out. In such approaches the system is described by a mixed state rather than a pure state: in quantum mechanics by a density instead of a wavefunction and in classical mechanics by a density function rather than a point in the phase space. There are other approaches that do deal with the evolution equations of a pure state. The canonical formulation of classical mechanics does not apply in a direct way to dissipative systems because the hamiltonian usually has the meaning of energy and would be conserved. -
An Ab Initio Definition of Life Pertaining to Astrobiology Ian Von Hegner
An ab initio definition of life pertaining to Astrobiology Ian von Hegner To cite this version: Ian von Hegner. An ab initio definition of life pertaining to Astrobiology. 2019. hal-02272413v2 HAL Id: hal-02272413 https://hal.archives-ouvertes.fr/hal-02272413v2 Preprint submitted on 15 Oct 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. HAL archives-ouvertes.fr | CCSD, August, 2019. An ab initio definition of life pertaining to Astrobiology Ian von Hegner Aarhus University Abstract Many definitions of life have been put forward in the course of time, but none have emerged to entirely encapsulate life. Putting forward an adequate definition is not a simple matter to do, despite many people seeming to believe they have an intuitive understanding of what is meant when it is stated that something is life. However, it is important to define life, because we ourselves, individually and collectively, are life, which entails an importance in itself. Furthermore, humankind‘s capability to look for life on other planets is steadily becoming a real possibility. But in order to realize that search, a definition of life is required. -
Dissipative Structures in Educational Change: Prigogine and the Academy
Wichita State University Libraries SOAR: Shocker Open Access Repository Donald L. Gilstrap University Libraries Dissipative Structures in Educational Change: Prigogine and the Academy Donald L. Gilstrap Wichita State University Citation Gilstrap, Donald L. 2007. Dissipative structures in educational change: Prigogine and the academy. -- International Journal of Leadership in Education: Theory and Practice; v.10 no.1, pp. 49-69. A post-print of this paper is posted in the Shocker Open Access Repository: http://soar.wichita.edu/handle/10057/6944 Dissipative structures in educational change: Prigogine and the academy Donald L. Gilstrap This article is an interpretive study of the theory of irreversible and dissipative systems process transformation of Nobel Prize winning physicist Ilya Prigogine and how it relates to the phenomenological study of leadership and organizational change in educational settings. Background analysis on the works of Prigogine is included as a foundation for human inquiry and metaphor generation for open, dissipative systems in educational settings. International case study research by contemporary systems and leadership theorists on dissipative structures theory has also been included to form the interpretive framework for exploring alternative models of leadership theory in far from equilibrium educational settings. Interpretive analysis explores the metaphorical significance, connectedness, and inference of dissipative systems and helps further our knowledge of human-centered transformations in schools and colleges. Introduction Over the course of the past 30 years, educational institutions have undergone dramatic shifts, leading teachers and administrators in new and sometimes competing directions. We are all too familiar with rising enrolments and continual setbacks in financial resources. Perhaps this has never been more evident, particularly in the USA, than with the recent changes in educational and economic policies. -
Entropy and Life
Entropy and life Research concerning the relationship between the thermodynamics textbook, states, after speaking about thermodynamic quantity entropy and the evolution of the laws of the physical world, that “there are none that life began around the turn of the 20th century. In 1910, are established on a firmer basis than the two general American historian Henry Adams printed and distributed propositions of Joule and Carnot; which constitute the to university libraries and history professors the small vol- fundamental laws of our subject.” McCulloch then goes ume A Letter to American Teachers of History proposing on to show that these two laws may be combined in a sin- a theory of history based on the second law of thermo- gle expression as follows: dynamics and on the principle of entropy.[1][2] The 1944 book What is Life? by Nobel-laureate physicist Erwin Schrödinger stimulated research in the field. In his book, Z dQ Schrödinger originally stated that life feeds on negative S = entropy, or negentropy as it is sometimes called, but in a τ later edition corrected himself in response to complaints and stated the true source is free energy. More recent where work has restricted the discussion to Gibbs free energy because biological processes on Earth normally occur at S = entropy a constant temperature and pressure, such as in the atmo- dQ = a differential amount of heat sphere or at the bottom of an ocean, but not across both passed into a thermodynamic sys- over short periods of time for individual organisms. tem τ = absolute temperature 1 Origin McCulloch then declares that the applications of these two laws, i.e. -
Perturbation Theory and Exact Solutions
PERTURBATION THEORY AND EXACT SOLUTIONS by J J. LODDER R|nhtdnn Report 76~96 DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS J J. LODOER ASSOCIATIE EURATOM-FOM Jun»»76 FOM-INST1TUUT VOOR PLASMAFYSICA RUNHUIZEN - JUTPHAAS - NEDERLAND DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS by JJ LODDER R^nhuizen Report 76-95 Thisworkwat performed at part of th«r«Mvchprogmmncof thcHMCiattofiafrccmentof EnratoniOTd th« Stichting voor FundtmenteelOiutereoek der Matctk" (FOM) wtihnnmcWMppoft from the Nederhmdie Organiutic voor Zuiver Wetemchap- pcigk Onderzoek (ZWO) and Evntom It it abo pabHtfMd w a the* of Ac Univenrty of Utrecht CONTENTS page SUMMARY iii I. INTRODUCTION 1 II. GENERALIZED FUNCTIONS DEFINED ON DISCONTINUOUS TEST FUNC TIONS AND THEIR FOURIER, LAPLACE, AND HILBERT TRANSFORMS 1. Introduction 4 2. Discontinuous test functions 5 3. Differentiation 7 4. Powers of x. The partie finie 10 5. Fourier transforms 16 6. Laplace transforms 20 7. Hubert transforms 20 8. Dispersion relations 21 III. PERTURBATION THEORY 1. Introduction 24 2. Arbitrary potential, momentum coupling 24 3. Dissipative equation of motion 31 4. Expectation values 32 5. Matrix elements, transition probabilities 33 6. Harmonic oscillator 36 7. Classical mechanics and quantum corrections 36 8. Discussion of the Pu strength function 38 IV. EXACTLY SOLVABLE MODELS FOR DISSIPATIVE MOTION 1. Introduction 40 2. General quadratic Kami1tonians 41 3. Differential equations 46 4. Classical mechanics and quantum corrections 49 5. Equation of motion for observables 51 V. SPECIAL QUADRATIC HAMILTONIANS 1. Introduction 53 2. Hamiltcnians with coordinate coupling 53 3. Double coordinate coupled Hamiltonians 62 4. Symmetric Hamiltonians 63 i page VI. DISCUSSION 1. Introduction 66 ?. -
Chapter 8 Stability Theory
Chapter 8 Stability theory We discuss properties of solutions of a first order two dimensional system, and stability theory for a special class of linear systems. We denote the independent variable by ‘t’ in place of ‘x’, and x,y denote dependent variables. Let I ⊆ R be an interval, and Ω ⊆ R2 be a domain. Let us consider the system dx = F (t, x, y), dt (8.1) dy = G(t, x, y), dt where the functions are defined on I × Ω, and are locally Lipschitz w.r.t. variable (x, y) ∈ Ω. Definition 8.1 (Autonomous system) A system of ODE having the form (8.1) is called an autonomous system if the functions F (t, x, y) and G(t, x, y) are constant w.r.t. variable t. That is, dx = F (x, y), dt (8.2) dy = G(x, y), dt Definition 8.2 A point (x0, y0) ∈ Ω is said to be a critical point of the autonomous system (8.2) if F (x0, y0) = G(x0, y0) = 0. (8.3) A critical point is also called an equilibrium point, a rest point. Definition 8.3 Let (x(t), y(t)) be a solution of a two-dimensional (planar) autonomous system (8.2). The trace of (x(t), y(t)) as t varies is a curve in the plane. This curve is called trajectory. Remark 8.4 (On solutions of autonomous systems) (i) Two different solutions may represent the same trajectory. For, (1) If (x1(t), y1(t)) defined on an interval J is a solution of the autonomous system (8.2), then the pair of functions (x2(t), y2(t)) defined by (x2(t), y2(t)) := (x1(t − s), y1(t − s)), for t ∈ s + J (8.4) is a solution on interval s + J, for every arbitrary but fixed s ∈ R. -
Role of Nonlinear Dynamics and Chaos in Applied Sciences
v.;.;.:.:.:.;.;.^ ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Oivisipn 2000 Please be aware that all of the Missing Pages in this document were originally blank pages BARC/2OOO/E/OO3 GOVERNMENT OF INDIA ATOMIC ENERGY COMMISSION ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Division BHABHA ATOMIC RESEARCH CENTRE MUMBAI, INDIA 2000 BARC/2000/E/003 BIBLIOGRAPHIC DESCRIPTION SHEET FOR TECHNICAL REPORT (as per IS : 9400 - 1980) 01 Security classification: Unclassified • 02 Distribution: External 03 Report status: New 04 Series: BARC External • 05 Report type: Technical Report 06 Report No. : BARC/2000/E/003 07 Part No. or Volume No. : 08 Contract No.: 10 Title and subtitle: Role of nonlinear dynamics and chaos in applied sciences 11 Collation: 111 p., figs., ills. 13 Project No. : 20 Personal authors): Quissan V. Lawande; Nirupam Maiti 21 Affiliation ofauthor(s): Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai 22 Corporate authoifs): Bhabha Atomic Research Centre, Mumbai - 400 085 23 Originating unit : Theoretical Physics Division, BARC, Mumbai 24 Sponsors) Name: Department of Atomic Energy Type: Government Contd...(ii) -l- 30 Date of submission: January 2000 31 Publication/Issue date: February 2000 40 Publisher/Distributor: Head, Library and Information Services Division, Bhabha Atomic Research Centre, Mumbai 42 Form of distribution: Hard copy 50 Language of text: English 51 Language of summary: English 52 No. of references: 40 refs. 53 Gives data on: Abstract: Nonlinear dynamics manifests itself in a number of phenomena in both laboratory and day to day dealings. -
ATTRACTORS: STRANGE and OTHERWISE Attractor - in Mathematics, an Attractor Is a Region of Phase Space That "Attracts" All Nearby Points As Time Passes
ATTRACTORS: STRANGE AND OTHERWISE Attractor - In mathematics, an attractor is a region of phase space that "attracts" all nearby points as time passes. That is, the changing values have a trajectory which moves across the phase space toward the attractor, like a ball rolling down a hilly landscape toward the valley it is attracted to. PHASE SPACE - imagine a standard graph with an x-y axis; it is a phase space. We plot the position of an x-y variable on the graph as a point. That single point summarizes all the information about x and y. If the values of x and/or y change systematically a series of points will plot as a curve or trajectory moving across the phase space. Phase space turns numbers into pictures. There are as many phase space dimensions as there are variables. The strange attractor below has 3 dimensions. LIMIT CYCLE (OR PERIODIC) ATTRACTOR STRANGE (OR COMPLEX) ATTRACTOR A system which repeats itself exactly, continuously, like A strange (or chaotic) attractor is one in which the - + a clock pendulum (left) Position (right) trajectory of the points circle around a region of phase space, but never exactly repeat their path. That is, they do have a predictable overall form, but the form is made up of unpredictable details. Velocity More important, the trajectory of nearby points diverge 0 rapidly reflecting sensitive dependence. Many different strange attractors exist, including the Lorenz, Julian, and Henon, each generated by a Velocity Y different equation. 0 Z The attractors + (right) exhibit fractal X geometry. Velocity 0 ATTRACTORS IN GENERAL We can generalize an attractor as any state toward which a Velocity system naturally evolves. -
Polarization Fields and Phase Space Densities in Storage Rings: Stroboscopic Averaging and the Ergodic Theorem
Physica D 234 (2007) 131–149 www.elsevier.com/locate/physd Polarization fields and phase space densities in storage rings: Stroboscopic averaging and the ergodic theorem✩ James A. Ellison∗, Klaus Heinemann Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, United States Received 1 May 2007; received in revised form 6 July 2007; accepted 9 July 2007 Available online 14 July 2007 Communicated by C.K.R.T. Jones Abstract A class of orbital motions with volume preserving flows and with vector fields periodic in the “time” parameter θ is defined. Spin motion coupled to the orbital dynamics is then defined, resulting in a class of spin–orbit motions which are important for storage rings. Phase space densities and polarization fields are introduced. It is important, in the context of storage rings, to understand the behavior of periodic polarization fields and phase space densities. Due to the 2π time periodicity of the spin–orbit equations of motion the polarization field, taken at a sequence of increasing time values θ,θ 2π,θ 4π,... , gives a sequence of polarization fields, called the stroboscopic sequence. We show, by using the + + Birkhoff ergodic theorem, that under very general conditions the Cesaro` averages of that sequence converge almost everywhere on phase space to a polarization field which is 2π-periodic in time. This fulfills the main aim of this paper in that it demonstrates that the tracking algorithm for stroboscopic averaging, encoded in the program SPRINT and used in the study of spin motion in storage rings, is mathematically well-founded. -
Phase Plane Methods
Chapter 10 Phase Plane Methods Contents 10.1 Planar Autonomous Systems . 680 10.2 Planar Constant Linear Systems . 694 10.3 Planar Almost Linear Systems . 705 10.4 Biological Models . 715 10.5 Mechanical Models . 730 Studied here are planar autonomous systems of differential equations. The topics: Planar Autonomous Systems: Phase Portraits, Stability. Planar Constant Linear Systems: Classification of isolated equilib- ria, Phase portraits. Planar Almost Linear Systems: Phase portraits, Nonlinear classi- fications of equilibria. Biological Models: Predator-prey models, Competition models, Survival of one species, Co-existence, Alligators, doomsday and extinction. Mechanical Models: Nonlinear spring-mass system, Soft and hard springs, Energy conservation, Phase plane and scenes. 680 Phase Plane Methods 10.1 Planar Autonomous Systems A set of two scalar differential equations of the form x0(t) = f(x(t); y(t)); (1) y0(t) = g(x(t); y(t)): is called a planar autonomous system. The term autonomous means self-governing, justified by the absence of the time variable t in the functions f(x; y), g(x; y). ! ! x(t) f(x; y) To obtain the vector form, let ~u(t) = , F~ (x; y) = y(t) g(x; y) and write (1) as the first order vector-matrix system d (2) ~u(t) = F~ (~u(t)): dt It is assumed that f, g are continuously differentiable in some region D in the xy-plane. This assumption makes F~ continuously differentiable in D and guarantees that Picard's existence-uniqueness theorem for initial d ~ value problems applies to the initial value problem dt ~u(t) = F (~u(t)), ~u(0) = ~u0. -
Calculus and Differential Equations II
Calculus and Differential Equations II MATH 250 B Linear systems of differential equations Linear systems of differential equations Calculus and Differential Equations II Second order autonomous linear systems We are mostly interested with2 × 2 first order autonomous systems of the form x0 = a x + b y y 0 = c x + d y where x and y are functions of t and a, b, c, and d are real constants. Such a system may be re-written in matrix form as d x x a b = M ; M = : dt y y c d The purpose of this section is to classify the dynamics of the solutions of the above system, in terms of the properties of the matrix M. Linear systems of differential equations Calculus and Differential Equations II Existence and uniqueness (general statement) Consider a linear system of the form dY = M(t)Y + F (t); dt where Y and F (t) are n × 1 column vectors, and M(t) is an n × n matrix whose entries may depend on t. Existence and uniqueness theorem: If the entries of the matrix M(t) and of the vector F (t) are continuous on some open interval I containing t0, then the initial value problem dY = M(t)Y + F (t); Y (t ) = Y dt 0 0 has a unique solution on I . In particular, this means that trajectories in the phase space do not cross. Linear systems of differential equations Calculus and Differential Equations II General solution The general solution to Y 0 = M(t)Y + F (t) reads Y (t) = C1 Y1(t) + C2 Y2(t) + ··· + Cn Yn(t) + Yp(t); = U(t) C + Yp(t); where 0 Yp(t) is a particular solution to Y = M(t)Y + F (t). -
Visualizing Quantum Mechanics in Phase Space
Visualizing quantum mechanics in phase space Heiko Baukea) and Noya Ruth Itzhak Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany (Dated: 17 January 2011) We examine the visualization of quantum mechanics in phase space by means of the Wigner function and the Wigner function flow as a complementary approach to illustrating quantum mechanics in configuration space by wave func- tions. The Wigner function formalism resembles the mathematical language of classical mechanics of non-interacting particles. Thus, it allows a more direct comparison between classical and quantum dynamical features. PACS numbers: 03.65.-w, 03.65.Ca Keywords: 1. Introduction 2. Visualizing quantum dynamics in position space or momentum Quantum mechanics is a corner stone of modern physics and technology. Virtually all processes that take place on an space atomar scale require a quantum mechanical description. For example, it is not possible to understand chemical reactionsor A (one-dimensional) quantum mechanical position space the characteristics of solid states without a sound knowledge wave function maps each point x on the position axis to a of quantum mechanics. Quantum mechanical effects are uti- time dependent complex number Ψ(x, t). Equivalently, one lized in many technical devices ranging from transistors and may consider the wave function Ψ˜ (p, t) in momentum space, Flash memory to tunneling microscopes and quantum cryp- which is given by a Fourier transform of Ψ(x, t), viz. tography, just to mention a few applications. Quantum effects, however, are not directly accessible to hu- 1 ixp/~ Ψ˜ (p, t) = Ψ(x, t)e− dx . (1) man senses, the mathematical formulations of quantum me- (2π~)1/2 chanics are abstract and its implications are often unintuitive in terms of classical physics.